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Consider that my question is not a personal and/or subjective question. Perhaps, you have hired a mathematics educator in your department and you are interested in finding a way to communicate with him/her and vice versa. Or not, for some reasons you are looking at one of the writings of those peoples, and more often than not you get disappointed since you can hardly find what you are looking for. What do you expect from mathematics education as a discipline?

Addendum: After being revised by others it occured to me that my description had the potential of being interpreted as rude or offensive. Thus I decided to accept the revision done. Furthermore, to be less argumentative , it would be great to include in your answers/comments some good examples of the ideas from mathematics education that you have found useful, and perhaps an explanation of how they were useful for you (there are one or two below). I hope this would be a posstive way to go.

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    $\begingroup$ I have voted to close this question as subjective and argumentative. If it is to be saved, at a minimum the first three sentences must go. I suppose that, being CW, I could just go in and change the question. Perhaps I will do so if the question is not closed, but even with the changes I see to make, I'm not sure I can make the question into an appropriate one for MO. $\endgroup$
    – Theo Johnson-Freyd
    Commented Apr 20, 2013 at 5:26
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    $\begingroup$ The question as it stands is unintelligible, and its original intention was purely social commentary. The discussion that's followed is entirely open-ended, highly opinionated, makes no pretense even of there being an attempt to find concrete solutions to concrete problems, and is only vaguely directed at the original post (necessarily so, as the original post doesn't actually make sense in the first place) . How has this not been closed yet? $\endgroup$
    – Amit Kumar Gupta
    Commented Apr 20, 2013 at 9:58
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    $\begingroup$ Unfortunately, MO is not intended for discussions, even those of great interest to mathematicians. $\endgroup$
    – Gerald Edgar
    Commented Apr 20, 2013 at 14:23
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    $\begingroup$ A more natural venue for your question would be the Mathematics Teaching Community website founded by (my colleague and departmental next-door neighbor) Sybilla Beckmann and (my PhD student) Jacob Hicks: see mathematicsteachingcommunity.math.uga.edu. I do advise you to phrase your question in a respectful (rather than "humorous") way there, in order to get the best response. $\endgroup$
    – Pete L. Clark
    Commented Apr 21, 2013 at 1:05
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    $\begingroup$ It seems to me that "mathematics education" denotes two different disciplines in universities. One is the effort to improve the teaching in the universities themselves, with training programs for teaching assistants and postdocs, classroom visits, etc. The other is teaching classes for students who are majoring in mathematics education (or majoring in education with an emphasis on mathematics) and who plan to teach at the elementary or secondary level. $\endgroup$
    – Andreas Blass
    Commented Apr 21, 2013 at 20:15

5 Answers 5

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I recall working with a reasonably reputable mathematics educator once, teaching a calculus class. At one point it became evident that the guy wasn't comfortable summing a geometric series. One thing I expect from mathematics education is that mathematics educators know some mathematics. Saying this any less bluntly would lose too much of the sentiment I'm trying to convey, so there it is.

As a mathematician, I'd personally like to see work in mathematics education that helps me teach what it is that I actually do as a mathematician. I was introduced by Ken Appel to some of the ideas of Hy Bass on mathematics education, e.g. the "granularity" concept which asserts that at different levels of sophistication mathematicians allow different jump sizes in their arguments. Awareness of granularity made explicit like this really has changed the way I organize my undergraduate course material and for me was revolutionary.

Other ideas of Bass that I'd like to see followed up include the idea of a common structure problem set. Such an idea might help to get a large chunk of a difficult aspect of the mathematical aesthetic into the curriculum.

In general, I'd like to see mathematics education address how the practices of the best mathematicians can be brought to the graduate and undergraduate population in universities, and how we can bring more of mathematics itself as experienced by mathematicians to our students. I'm more interested in this than this than studies of how to improve calculus course assessment, for example. I've always been frustrated that nobody seems to study the learning approaches of successful mathematicians rather than average students. I'd personally like to see more of our best practices being studied and propagated. This last paragraph is a bit ignorant, I know, but it's my honest impression.

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    $\begingroup$ Would you be interested in a resurgence of the somewhat reviled "new math"? This talk outlines the history a bit: lsc-net.terc.edu/do/conference_material/6857/show/… $\endgroup$
    – S. Carnahan ♦
    Commented Apr 20, 2013 at 3:12
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    $\begingroup$ What would you suggest as a good starting place for learning about Bass's ideas on granularity? $\endgroup$
    – Henry Cohn
    Commented Apr 20, 2013 at 3:59
  • $\begingroup$ @Henry: To my chagrin, I only find mention of it in the paper to which I linked above as "grain size". Maybe that's how it is referred to in the literature. All I know is what I wrote above, and that was enough to make a difference for me. If anyone has a source, I'd love to have it, too. $\endgroup$
    – Jon Bannon
    Commented Apr 20, 2013 at 10:37
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    $\begingroup$ @S.Carnahan: I knew someone would bring this up! Implicit in my answer is an earlier comment I deleted: Personally, I'm not as interested in educating the general populace as I am in educating undergraduate and graduate students. (This isn't something I'm proud of!) So I'm not interested in a renewal of the "new math". $\endgroup$
    – Jon Bannon
    Commented Apr 20, 2013 at 10:39
  • $\begingroup$ @Scott: great link. $\endgroup$
    – Steve
    Commented Apr 20, 2013 at 18:45
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To answer your question truthfully, I do not think most mathematicians expect anything from mathematics education as a discipline. It is not even on the radar.

The vast majority of mathematicians have never had any formal training as teachers at all. When they get jobs as educators, there is no kind of systematic support for teaching. There might be services available if you seek them out, but basically the good teachers are all ready good, and the bad teachers generally do not care enough to seek out help.

I have no idea how to change this, or even whether changing this would be a good thing. In my experience the vast majority of mathematics education research is essentially at the level of anecdotes. I think that meaningful education research is only just now starting to be possible, with the kind of huge data sets we can now gather on students taking courses on computers.

If you personally want to improve mathematics education at your university, then probably you should try to make friends with as many mathematicians as possible, and see if you could personally work with them to change their teaching habits. Try to make it fun.

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    $\begingroup$ I did not downvote and have no personal problem with it but this sentence (in its generality and vagueness, in particular) "In my experience the vast majority of mathematics education research is essentially at the level of anecdotes." is something I could well find somebody seeing as problematic. $\endgroup$
    – user9072
    Commented Apr 19, 2013 at 20:46
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    $\begingroup$ Your penultimate paragraph makes me wonder: (1) What is your "experience" that has led you to believe "the vast majority of mathematics education research is essentially at the level of anecdotes"? (2) Why do you think that "meaningful education research" is only now possible with "huge data sets" and online course-taking? $\endgroup$
    – Benjamin Dickman
    Commented Apr 19, 2013 at 20:46
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    $\begingroup$ @Benjamin - I have read a lot of papers in mathematics education. I thought about getting a Ph.D. in mathematics education at one time. It is not an objective science, and there are huge culture wars in the field. For example, between constructivists and people who encourage rote memorization. Read en.wikipedia.org/wiki/… for example, including the criticism section. $\endgroup$
    – Steve
    Commented Apr 19, 2013 at 20:53
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    $\begingroup$ For example, think about teaching double digit arithmetic. There are a lot of mathematics educators who will tell you that teaching the algorithm is essentially killing any chance that the student will understand it. There are others who say that you have to learn the mechanics first, and then you have the structure to start trying to understand what is going on. Put these two people in a room, give them access to the whole internet worth of studies, and there is basically no chance that either person will budge an inch. $\endgroup$
    – Steve
    Commented Apr 19, 2013 at 20:59
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    $\begingroup$ I would have voted this up, but the third paragraph was a bit of a buzzkill. If anything, our increasing tilt toward numerical data in education reform is causing us to focus on quantifiable-but-noisy-and-unimportant things like standardized tests, at the expense of important-but-inherently-fuzzy things like classroom best practices. Also, I think the notion of "objective science" is incredibly slippery. Even though we mathematicians have mostly agreed on what constitutes a proof, hiring and journal acceptance processes are far from objective, and they strongly influence future research. $\endgroup$
    – S. Carnahan ♦
    Commented Apr 20, 2013 at 16:33
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I think the expectation is that math educators at a math department will be leaders on innovative ways to teach and assess students' progress, will train and mentor TAs, take part in and initiate pilot projects, and help with technology training. (Math and non-math) educators at my institution are excellent, and do all the above. Like everything else, teaching gets better with practice and faculty and TAs could sure benefit from professional advice.

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Steve's answer suggests that maybe another question should be asked, prior to Amir's: What do mathematicians think mathematics educators do? Am I right?

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You might want to take a look at the March 2011 issue of the Notices of the American Mathematical Society. It's a special issue on mathematics education.

The articles herein explore different means by which research mathematicians can get involved in the education process.

The articles are about curriculum development, teacher preparation, certifying teacher content knowledge, and student participation in mathematics competitions.

Intentionally absent are discussions on "squabbling, on intellectual, political, and even financial levels, over who knows best" and on the role of the mathematician as corrector of errors in textbooks.

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